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## The spectral radius of a simple graph

I’ve decided not to port over the posts from AoPS. Among other things, I’d prefer to preserve the comments on older posts as they are.

So here’s something I’ve been thinking about. Given a simple connected graph $G$, define $\rho(G)$ to be its largest positive eigenvalue; in other words, if $w_n$ counts the number of walks (equivalently, closed walks) of length $n$ on $G$, then

$\displaystyle \rho(G) = \limsup_{n \to \infty} \sqrt[n]{w_n}.$